Membership Problem in groups acting freely on Zn-trees

Groups acting freely on Zn-trees (Zn-free groups) play a key role in the study of non-archimedean group actions. Following Stallingsʼ ideas, we develop graph-theoretic techniques to investigate subgroup structure of Zn-free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in Zn-free groups.